We used collapsible water carriers when we came to BM in 2000. We had a bunch of 20-litre ones and a bunch of 10-litre ones. Like you describe, we were tetris-packed into our van. We drove from Winnipeg and filled the containers in Reno. We were able to give up that extra space for a few hours on the way in but certainly would not have been able to do so for the full length of the drive (2 days each way).
The brand we used is called Reliance Fold-A-Carrier. Not sure if they're still available or if the quality is as good as it was 12 years ago, but we kept a few of ours and still occasionally use them when we go camping, and we've had no problems with them. Looking at reviews, I see complaints about them leaking with heavy/frequent use. Other people report "pinhole leaks" after "many years of use/folding," and still others say they "last forever." A week at Burning Man should be fine so long as you're relatively gentle with them (and my experience suggests that this is true).
The only complaint I have about them is that the handle is insanely uncomfortable. When you're carrying the big one full of water, the plastic corners are very hard in the palms of your hands. While camping last weekend, I found myself wishing I had work gloves when I had to haul it from the water tap.
If you want drama to stop following you everywhere, try letting go of the leash.